Cs2Fe2(MoO4)3—A Strongly Frustrated Magnet with Orbital Degrees of Freedom and Magnetocaloric Properties

We report an in-depth study of the thermodynamic and magnetocaloric properties of a strongly frustrated magnet, Cs2Fe2(MoO4)3. The underlying structure belongs to the double trillium lattice, which consists of two FeII (S = 2) sites with easy-axis and easy-plane single-ion anisotropy. Detailed 57Fe Mössbauer spectroscopic investigations along with ligand-field calculations support the existence of disparate ground states. The antiferromagnetic ordered structure is presented by the propagation vector k = (0,0,0) with noncollinear magnetic moments of 2.97 μB (Fe1) and 0.17 μB (Fe2), respectively. Strong and disordered magnetic correlations exist in the temperature regime between TN ≈ 1.0 K and |θCW| ≈ 22 K. The large degeneracy of the ground state is investigated in terms of its magnetocaloric response. Magnetization and specific heat measurements indicate a significant magnetocaloric cooling efficiency, making this rare-earth-free compound a promising candidate for cryogenic magnetic refrigeration applications, with refrigeration capacity of 79 J kg–1 for Δ(μ0H) = 8 T.

S1. Rietveld refinement of powder X-ray diffraction data for Cs2Fe2(MoO4)3 at room temperature.Bragg positions are represented by bars, the difference between data (black dots) and the fit (green line) is given by a gray line.

NEUTRON DIFFRACTION DATA
Neutron diffraction data were collected at the HB-2A beamline at ORNL (proposal number IPTS-31426.1) and refined using the Fullprof-suite [2].Crystallographic data for 10 K are provided in Tab.S2.Pronounced diffuse scattering, indicative of short-range ordering, is observed below 10 K and persists with minimal change down to 2 K, see Fig. S2 for difference in intensities ∆(Int.) at selected temperatures.In Fig. S3 we show the Rietveld refinement of data below T N .Details of the refinement below 1.1 K are given along with the magnetic moments of individual Fe-sites in Tab.S3.Information on the magnetic space group is provided in Tab.S4, supported by the magnetic CIF format (magCIF).The development of LRO below 1.0 K based on normalized intensities is shown in Fig. S4.

AOM CALCULATIONS
Graphical schemes provide an overview of low-energy states for Fe II ions in ligand fields of D 3d and C 3v symmetry including spin-orbit coupling, Fig. S5.Fig. S6 illustrates the field dependence of the lowest energy levels.The calculated temperature-dependent magnetic susceptibility for both ligand fields is provided in Fig. S7, which focuses on the single-ion anisotropy in the purely paramagnetic state.) )

DIPOLE-DIPOLE INTERACTION
An order-of-magnitude estimate of the magnetic dipole-dipole interaction between the Fe II ions in the paramagnetic state is provided based on the AOM calculation and the room-temperature structure in Tab.S5.
We calculate the order-of-magnitude estimate of the energy for magnetic dipole-dipole interaction between the Fe II magnetic moments by the approximation of fixed permanent dipoles with freely rotating dipolar moments.These are thermally averaged over all angles (paramagnetic analog of the Keesom interaction [3]).For the calculation, we used the distances between Fe II ions from the room-temperature structure data (Tab. 1 in the main text) and the effective magnetic moments obtained from AOM calculations (Tab.S5) set into the following equation: (S1) Here, w dd,A denotes the energy of the dipole-dipole interaction of Fe II at the selected site (A = Fe1 or Fe2), with the effective magnetic moment µ A , and other Fe II i ions ("B-site") at a distance r Ai , which have the effective magnetic moment of µ i .The sum runs over all nearest and next-nearest Fe II ions up to a distance of 7 Å (in total 13 neighbors); µ 0 is the permeability of vacuum.The overview of calculated energy w dd,A for individual Fe sites and selected temperatures is provided in Tab.S5.
While this approximation is more precise at higher temperatures, at which the system is in a pure paramagnetic regime with diminished spin-orbit-coupling anisotropy, it still provides a reasonable estimate of the energy scale.Importantly, in the static case, the magnetic dipole-dipole interaction between the nearest neighbors practically vanishes due to their different single-ion anisotropies, which would render their magnetic moments approximately perpendicular.TABLE S5.Effective magnetic moments of Fe II ions in Fe1 and Fe2 sites obtained from AOM calculations, and the calculated effective magnetic dipole-dipole interaction according to Eq. S1 at selected temperatures.The transmission Mössbauer spectra of Cs 2 Fe 2 (MoO 4 ) 3 were analyzed with the Recoil software [4] by using the Extended Voigt-based Fitting (xVBF) analysis [5] to extract the Gaussian distribution of the quadrupole splitting and hyperfine magnetic field for each site.Representative fits of the 57 Fe Mössbauer spectra for the lowest and highest measured temperatures are shown in Fig. S8 and the parameters obtained from the fits are summarized in Tab.S6.The full width at half maximum of the Lorentzian natural line was fixed to 0.28 mm s -1 .For temperatures below 22 K, the Fe1 subspectrum was fitted by two independent components with a fixed center shift, δ exp , to obtain the information on the fluctuating hyperfine field, H hf , as well as quadrupole splitting, QS.At higher temperatures, both Fe1a and Fe1b subspectra were linked as subcomponents of a quadrupole-split component with identical δ exp .The provided errors of hyperfine parameters obtained from the fits were calculated by using Monte Carlo sampling with counting statistics and correspond to the 95% confidence interval.The absence of any discontinuity in the hyperfine parameters of Fe1 and Fe2 subspectra excludes structural phase transitions.The intensity of a spectral component I is directly proportional to the Lamb-Mössbauer (recoil-free) factor f of the respective site, whose temperature dependence also determines I(T ).Similarly to Debye-Waller factor in XRD, the Lamb-Mössbauer factor is related to the mean square displacement of the nucleus from its equilibrium position, ⟨x 2 ⟩, in the direction of propagation of γ-rays [6].The intensities of individual spectral components were calculated from the spectral area A normalized by the respective background intensity I bg as I = A/I bg and are depicted in Fig. S9A.The center shift is composed of two contributions, the isomer shift, δ 0 , which results from the electric monopole interaction of the 57 Fe nucleus and its surroundings, and the second-order Doppler shift, δ SOD .The δ SOD shift originates in a relativistic change in energy of the γ-photon due to thermal vibrations of the 57 Fe nuclei, and is proportional to the mean square velocity ⟨v 2 ⟩ (or mean kinetic energy) of the 57 Fe atom.From the temperature dependence of δ SOD , we determine the Mössbauer temperature, θ M , an analog of the Debye temperature specific for the Fe nuclei in their environment.The fits of δ exp (T ) to Eq. S2 [6,7] are depicted in Fig. S9B.
where E γ = 14.41 keV is the energy of the Mössbauer transition, M eff is the effective mass of the vibrating 57 Fe nuclei (fixed to the isotopic mass of 57 Fe, 53095 MeV c −2 , to stabilize the fit).We obtain δ 0 (Fe1) = 1.40 mm s -1 , θ M (Fe1) = 404(4) K and δ 0 (Fe2) = 1.38 mm s -1 , θ M (Fe2) = 308(8) K.The θ M values evidence the disparate ground states of the two Fe-sites.HEAT CAPACITY DATA Details on the fitting procedure of the heat capacity by the Debye-Einstein integral to extract the magnetic heat capacity are described, the parameters are listed in Tab.S7.The fitted curve of the phonon contribution along with the experimental data and an isotypic non-magnetic compound Cs 2 Mg 2 (MoO 4 ) 3 is illustrated in Fig. S10.The extracted magnetic heat capacity C m /R of Cs 2 Fe 2 (MoO 4 ) 3 and its low-temperature fits are presented in Fig. S11.The simulations of magnetic heat capacity corresponding to Schottky anomalies based on lowest energy levels of the two Fe sites and the corresponding magnetic entropy change is compared with experimental values in Fig. S12 for different applied magnetic fields.
The experimental heat capacity measured at zero field was fitted by the Debye-Einstein integral to approximate the phonon contribution: in which the Debye integral is defined as and the Einstein terms by E(y) = R y 2 e y (e y − 1) 2 , where y = In these equations, θ D and θ E denote the Debye and Einstein temperatures, R the gas constant, m and n i are fitting parameters approximating the number of the respective modes, whose sum m + i n i is equal three-times the number of atoms in the formula unit (i.e., 57 in our case).In the fitting procedure, all three acoustic -Debye -modes are approximated by a single Debye temperature (θ D1 , D 1 group with m = 3).The remaining optical modes are clustered into four groups (j = 4) described by four Einstein temperatures (θ Ei ).To reduce the number of free parameters, the anharmonic corrections (see, e.g., [8,9]) were not applied.For consistency, the fitting parameters were compared to other Cs-containing molybdates, such as Cs 2 Mo 2 O 7 [10] and Cs 2 Mo 2 O 7 • CsX (X = Cl, I) [11].These provide an estimate for the high-energy Einstein modes, which correspond primarily to fundamental stretching vibrations (E 4 ) and bending modes (E 3 ) of the molybdate, while E 1 is dominated by the heavy Cs + counterions.We used the diamagnetic isotypic Cs 2 Mg 2 (MoO 4 ) 3 compound [12] to derive the Debye temperature (θ D1 ) from a fit of C p /T vs. T 2 plot at low temperatures.The model parameters obtained from fits to the Debye-Einstein integral for this compound are provided in Tab.S7.A simple subtraction of neither raw nor re-scaled C p (T )/T data for the Mg-compound provide a reasonable estimate of the magnetic heat capacity of Cs 2 Fe 2 (MoO 4 ) 3 , see Fig. S10.Mg II and Fe II differ (i) in atomic weight, (ii) with respect to their ionic radii in six-fold coordination (0.72 Å for Mg II and 0.78 Å for high-spin Fe II , [13]), and (iii) in terms of bonding (ionic vs. coordinate).In order to model the phonon part of the specific heat for Cs 2 Fe 2 (MoO 4 ) 3 , we fixed θ D1 , θ E1 , θ E4 and the corresponding numbers of modes to the respective values of the Mg-compound.Then we fitted parameters of E 2 and E 3 modes, in which the vibrations within the [FeO 6 ] polyhedra are dominantly involved (see also Mössbauer temperatures θ M for the two iron sites).Note that while the Einstein temperatures do not differ significantly, the redistribution of mode numbers indicates the differences in bonding for the [FeO 6 ] complex.The final fitting parameters obtained for Cs 2 Fe 2 (MoO 4 ) 3 in zero applied field are summarized in Tab.S7 and were used to apply the phonon correction for C p (T ) to extract the magnetic contribution, C m (T ).The magnetic heat capacity C m (T ) was fitted in the temperature range of 2-7 K by a power law and a model simulating an energy gap of ∆E g , see Fig. S11B and Tab.S8.At zero applied field and at 1 T, the power law provided exponents of α(0 T) = 1.074 (8) and α(1 T) = 1.024 (6).Data obtained in applied magnetic fields of 3 T could be perceived at the transition between the two models.For higher fields, the data are described by the exponential fit, providing the energy gap of ∆E g .Next we compare experimental C m /T (T ) data in applied magnetic fields with the calculated magnetic heat capacity based on a three-level Schottky model [14,15]: Here, E i refer to the Fe-site specific energies (given in Fig. S6) and Z is the partition function.We show the averaged value, C m (T )/T, over the two Fe-sites in Fig S12 .We conclude that the magnetic contribution to the experimental heat capacity is not explained merely by Schottky-like anomalies of non-interacting (purely paramagnetic) Fe II -ions.
. Experimental (empty symbols) and calculated (lines) magnetic heat capacity C m/T. Calculations are based on a three-level Schottky model (Eq.S6) for each Fe-site and then averaged to conform with Cm/T in units per mol Fe.

MAGNETIZATION DATA
Comparison of DC magnetic susceptibility measured in zero-field-cooled and field-cooled modes in selected fields and the reciprocal susceptibility with Curie-Weiss fits are presented in Fig. S13.We used χ −1 = [C/(T − θ CW ) + χ TIP ] −1 , where temperature-independent paramagnetism was fitted at µ 0 H = 10 mT (χ TIP ≈ 7•10 -3 emu mol -1 ) and 100 mT (χ TIP ≈ 2•10 -3 emu mol -1 ) for a fixed θ CW = -22 K.At high temperatures the effective exchange can be approximated by a mean field approach (S7): with the multiplicity given in Tab.1 (main text).Using S = 2 for both Fe-sites gives J eff = -11 K (AFM).This is reasonably close to the temperature adjustment given for the simulation of the magnetization data at 100 K (Fig. S14 left panel).At 10 K Fe1 (S = 2) and Fe2 (S ′ = 1) holds for which we give averaged calculated magnetization (blue dashed lines) based on the respective Brillouin functions.Simulation of the experimental data at selected temperatures are provided as dotted lines with T 2 referring to the adjusted Brillouin function for S = 2 and T 1 in reference to S = 1 (Fig. S14 middle and right panel).m 0 H ( T )

FIG. S14
. Experimental field dependent magnetization data per Fe (black circles) for Cs2Fe2(MoO4)3 at selected temperatures.Calculated magnetization curves (g eff = 2.27) using Brillouin functions (B.F.) for paramagnetic S = 2 (brown line), S = 1 (red line), and the mixed spin state scenario (dashed blue line) are given for reference at experimental temperatures.Simulations (dotted lines) refer to the sum of the magnetization derived from the respective B.F.'s for S = 2 and S = 1 at temperatures T2 and T1, respectively.
The calculation of magnetocaloric parameters is based on the field-dependent magnetization curves measured at various temperatures (Fig. S15).

MAGNETOCALORIC EFFECT
Relevant to the magnetocaloric characteristics of a material are the magnetic entropy change −∆S m (T ) and the adiabatic temperature change ∆T ad , which are displayed in Fig. S16 for zero initial field and various final magnetic fields.The field-dependent magnetic entropy change isotherms were fitted at low-and high-field regions to a power law, whose temperature-dependent exponent n is shown in Fig. S17.The exponent n obtained from the fits of field-dependent ∆S m isotherms by the power law ∆S m(H,T ) = a + b(µ0H ) n(T ) at (i) low magnetic fields in the gapless region and at (ii) high fields in the gapped region (see the main text).

2 Q
FIG.S1.Rietveld refinement of powder X-ray diffraction data for Cs2Fe2(MoO4)3 at room temperature.Bragg positions are represented by bars, the difference between data (black dots) and the fit (green line) is given by a gray line.
FIG. S2.Diffuse scattering of powder neutron diffraction data (6 h scans, λ = 2.406 Å, HB-2A) for Cs2Fe2(MoO4)3.∆Int.data are obtained by subtracting the respective diffraction patterns for selected temperatures.Horizontal lines refer to the baselines of the shifted data, respectively.

FIG
FIG. S5.AOM calculations based on the room temperature crystal structure.(A) Splitting of the 5 T 2g ground state (O h symmetry) under trigonal distortion (elongation/compression) of the [FeO6] polyhedron along the three-fold axis (D 3d ).(B) Displacement of Fe-ions along the [111] axis breaks inversion symmetry (C 3v symmetry).(C) Energy level diagrams including spin-orbit coupling.We grouped the calculated energies into three levels with averaged energies (∆1,2) referring to the fits of QS (T ), see main text.The blue shaded graph illustrates the angular dependence of ∆1,2 for the degenerate case (Fe1).
FIG. S7.Calculated (AOM) magnetic susceptibility of paramagnetic Fe-sites in Cs2Fe2(MoO4)3 with χ ∥ along the ⟨111⟩ axis.The anisotropy is shown in gray as χ ∥ −χ ⊥ .Easy axis for Fe1 and easy plane anisotropy for Fe2.(Right) Single-ion anisotropies of the individual sites within the nn dimer.Ellipsoids represent the calculated direction and magnitude of the Fe II magnetic moments at 4 K.

Representative spectra at 4 K
and 250 K are shown in Fig. S8.The temperature dependence of line intensities and center shift are plotted in Fig. S9.Hyperfine parameters for 57 Fe Mössbauer spectra of Cs 2 Fe 2 (MoO 4 ) 3 are given in Tab.S6.
FIG. S8.Transmission 57 Fe Mössbauer spectra of Cs2Fe2(MoO4)3 at (A) 4 K and (B) 250 K showing the individual spectral components.The bottom green line gives the difference between the fit and experimental data.

) 3 .
For each component, δexp denotes the center (isomer) shift, QS and σQS the average and the Gaussian width of the quadrupole splitting distribution, H hf the effective magnetic hyperfine field, A rel the relative ratio of the site in Cs2Fe2(MoO4)3 (excluding Fe in the Be window) obtained from the spectral area A, and I is the spectral intensity.The change in models at 22 K used for the Fe1 site is marked by the horizontal line.(mm s -1 ) (mm s -1 FIG. S10.Molar heat capacity of Cs2Fe2(MoO4)3, C p/T is shown per formula unit in various applied fields, along with the isotypic Mg-compound data for comparison.Fits are represented as lines.
FIG. S11.Magnetic heat capacity, obtained by subtraction of the fitted phonon contribution (A) and low-temperature fits (B) provided per mol Fe.
FIG. S15.Specific magnetization of Cs2Fe2(MoO4)3, employed in the calculation of magnetocaloric parameters.Temperaturedependent magnetization at various fields between 0 and 8 T (left panel), derived from the curves shown in the right panel (lines at 0.05 T increments).Field-dependent scans measured at various temperatures; the ripples in the measured data around M ≊ 5 A m 2 kg -1 and µ0H ≊ 2.75 T are of instrumental origin and relate to change-of-sensitivity and touchdown-centering procedures, respectively.
FIG. S16.Magnetocaloric parameters of Cs2Fe2(MoO4)3.(A)The total isothermal magnetic entropy change derived from M (H ) isotherms and (B) the total adiabatic temperature change for various differences between the initial and final magnetic fields, with the initial field of 0 T.

m 0
FIG. S17.The exponent n obtained from the fits of field-dependent ∆S m isotherms by the power law ∆S m(H,T ) = a + b(µ0H ) n(T ) at (i) low magnetic fields in the gapless region and at (ii) high fields in the gapped region (see the main text).

TABLE S4 .
Magnetic space group (MSG) for Cs2Fe2(MoO4)3 with basic information about its relation with its parent paramagnetic structure.

TABLE S6 .
Parameters obtained from the xVBF fit of 57 Fe Mössbauer spectra of Cs2Fe2(MoO4

TABLE S7 .
Model parameters -Debye and Einstein temperatures and numbers of the respective modes -used for fitting zero-field heat capacity by Eq.S3 between 50 and 150 K for Cs2Fe2(MoO4)3 and Cs2Mg2(MoO4)3 (fitted range 3-280 K).Fixed parameters are marked by the asterisk, the errors provided are a rough estimate.